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Munich Personal RePEc Archive
Less is More: Capital Theory and
Almost Irregular-Uncontrollable Actual
Economies
Mariolis, Theodore and Tsoulfidis, Lefteris
Department of Public Administration, Panteion University,
Department of Economics, University of Macedonia
27 January 2018
Online at https://mpra.ub.uni-muenchen.de/84214/
MPRA Paper No. 84214, posted 27 Jan 2018 17:06 UTC
1
Less is More: Capital Theory and Almost Irregular-Uncontrollable
Actual Economies*
THEODORE MARIOLIS1 & LEFTERIS TSOULFIDIS2
1 Department of Public Administration, Panteion University, 136 Syngrou Ave, 17671
Athens, Greece, E-mail: [email protected] 2 Department of Economics, University of Macedonia, 156 Egnatia Str., 54006
Thessaloniki, Greece, E-mail: [email protected]
Capital theory and the associated with it price effects consequent upon changes in the
distributive variables hold centre stage when it comes to the internal consistency of both
classical and neoclassical theories of value. This paper briefly reviews the literature and
then focuses on the detected skew eigenvalue distribution of the vertically integrated
technical coefficients matrices of actual economies. The findings prompt the use of the
Schur triangularization theorem for the construction even of a single industry from the
input-output structure of the entire economy. Such a hyper-basic industry, in
combination with hyper-non-basic industries, embodies properties that may capture the
behaviour of the entire economic system. Thus, we can derive some meaningful results
consistent with the available empirical evidence, which finally suggest that actual
economies tend to respond as ‘irregular-uncontrollable’ systems.
Key words: Almost irregular-uncontrollable economy, Capital theory, Eigenvalue
distribution, Hyper-basic industry, Effective rank
JEL classifications: B21, B51, C67, D46, D57
1. Introduction
One of the enduring, puzzling and still not from the fully resolved issues in economic
theory is the effects of changes in income distribution on commodity prices. We know
that Ricardo ([1821] 1951, pp. 30-43) was from the first to formulate the question and to
argue that a definitive answer can be only obtained with the possession of an “invariable
measure of value”. That is, a commodity whose value would, under all technological and
distributional circumstances, remain the same and using this as the numéraire
commodity, we could identify the source of changes in the prices of all other
commodities. Ricardo devoted in vain his entire intellectual life to defining either
* We thank, without implicating, Scott Carter and Anwar Shaikh for their helpful comments in an earlier
version of this paper presented at the URPE panels of the Eastern Economic Association meeting, New
York, February 2017.
2
analytically or practically such a standard of value, which would remain invariant to
both changes in income distribution and production conditions. Marx ([1894] 1959,
Chap. 11) also faced a similar problem and proposed a solution on the basis of the
difference of an industry’s capital-intensity from the economy-wide average capital-
intensity.
The advent of neoclassical theory at the end of the nineteenth century defined
relative prices as indexes of relative scarcity. As a consequence, the prices of factors of
production were theorized to move monotonically in the upward or downward direction
with changes in income distribution. However, the determination of the price of a unit of
capital in a way which would be consistent with the premises of the neoclassical theory
was very hard to pinpoint. Robinson (1953) inspired by Piero Sraffa’s teaching and
writings exposed the inconsistencies in the neoclassical theorization of capital as a
‘factor of production’. Subsequently, Sraffa (1960) changed fundamentally the
established ideas on the relations between commodity prices and income distribution.
The underlying idea in Sraffa’s (1960) analysis is that an industry’s capital-
intensity depends on changes in income distribution which may initiate complex
movements in relative prices that may even alternate the characterization of an industry
from capital to labour intensive and vice versa. Consequently, the old classical rule
according to which the change in relative prices is strictly related to the capital-intensity
of the industry relative to others or some kind of invariable average does not in general
hold. Furthermore, Sraffa showed that it is possible that a capital-intensive technique
may be chosen for both low and high rates of profit, a result that runs contrary to the
neoclassical theory of scarcity prices. Under these circumstances, the determination of a
well-behaved demand for capital schedule is in question, and if such a core schedule is
questioned, then the presence of interdependency rules out the possibility of confidently
determining the remaining important demand and supply schedules. The consequences
for neoclassical analysis are thus quite upsetting (Tsoulfidis, 2010, p. 207).
Thus, the famous Cambridge capital controversies of the 1960s and 1970s have
shown that long-period prices do not necessarily display monotonic paths with respect to
changes in income distribution; as a consequence, the profit rate could not be taken as a
consistent index of the relative scarcity of capital. These theoretical findings, however,
were not corroborated by analogous empirical evidence, either because such research
was extremely difficult to pursue at that time, or for the reason that, if a theory is found
logically inconsistent, then there is no any pressing reason to test it empirically.
3
In each case, those controversies were conducted on purely theoretical grounds
without necessarily having any contact with actual economic data, as this can be judged
by the numerical examples utilized on both sides of the debate. For instance, on the US
side Cambridge, there is Samuelson’s (1962) parable of a one-commodity world, the
associated with it strictly linear wage-profit rate (WPR) curves and the well-behaved
supply of capital schedules. And this was sharply contrasted to the multi-commodity-
world of the UK side Cambridge Sraffian economists, whose numerical examples and
WPR curves are characterized by ‘any’ number of curvatures and possible shapes of
supply of capital schedules. These ‘exotic’ shapes of the WPR curves indicate that the
price-profit rate (PPR) curves can display extremes and inflection points rendering
untenable the neoclassical theorization of prices as scarcity indexes and proving that the
capital-intensity could not be defined in any uncontroversial way.
Summing-up, the theoretical findings were more in favour of the Sraffa-inspired
critique as Samuelson (1966), the leading figure from the neoclassical camp, admitted.
The same is true with Robert M. Solow and Charles E. Ferguson, while the list could be
extended to include Lucas (1988), who opined that the debate was won from the
Cambridge UK side and so did Mas Colell (1989), by noting that the relationship
between capital-intensity and profit rate could take ‘any’ possible shape. Although major
neoclassical economists have admitted the weakness of their theory to come to terms
with the results of the capital theory controversies, they, however, characterized them
‘paradoxical’, in the sense that these results contradict with the widely accepted
principles of neoclassical theory which are put on par with ‘common sense’. This
characterization might be justifiable, at least partly, for phenomena that may be
exempted from the ‘law of consumer’s demand’, such as, for instance, the well-known
‘Giffen paradox’. But, it is hard to accept it for those that arise in international trade and
came to be known as ‘Leontief paradox’ (see, e.g. Metcalfe and Steedman, 1979;
Paraskevopoulou et al., 2016) and much more difficult to accept it for those findings of
the capital theory that undermine the core propositions of the neoclassical theory. It
seems, however, that, despite these serious unsolved complications, the neoclassical
economists gradually lost interest in the capital theory debates, while the newer
generations of neoclassical economists rarely refer to these issues and continue using
various forms of ‘production functions’ as if there was no problem with the theory they
are based on.
4
In the meantime, the production price-wage-profit rate system of actual economies
(but, ex hypothesis, linear, closed and single-product) has been examined in a relatively
large number of studies. From Sekerka et al. (1970), Krelle (1977) and Shaikh (1984,
1998) onwards, the key stylized findings in these empirical studies are that:
(i). The vectors of vertically integrated labour coefficients, or labour values, and ‘actual
production prices’ are close to each other, as judged by alternative measures of
deviation.1 The estimated deviations are not too sensitive to the type of measure used for
their evaluation (Mariolis and Tsoulfidis, 2010, 2014a; Mariolis, 2011; Mariolis and
Soklis, 2011).
(ii). The ‘actual profit rate’ is usually no greater than 50% of its maximum feasible value
and, most of the time, is in the range of 30% to 40%. Therefore, the polynomial
approximation (Steedman, 1999) of the actual production prices, expressed in terms of
Sraffa’s (1960, Chaps. 4-5) Standard commodity (SSC), through ‘dated quantities of
embodied labour’ requires the inclusion of just a few terms (Tsoulfidis and Mariolis,
2007).
(iii). Non-monotonic PPR curves, expressed in terms of SSC, are not only relatively rare
(i.e. not significantly more than 20% of the tested cases) but also have no more than one
extreme point. Cases of reversal in the direction of deviation between production prices
and labour values (‘price-labour value reversals’) are rarer. In fact, the price-movement
is, more often than not, governed by the ‘capital-intensity effect’, i.e. by the difference
between the industry’s vertically integrated capital-intensity and the capital-intensity of
the Sraffian Standard system (SSS), where the latter equals the reciprocal of the
maximum feasible value of the profit rate. However, this ‘traditional flavour’ condition
can be modified by the ‘price effect’, i.e. the revaluation of the industry’s vertically
integrated capital, which depends on the entire economic system and, therefore, is not
predictable at the level of any single industry (Sraffa 1960, pp. 14-15; Pasinetti 1977, pp.
82-84; Mariolis et al. 2015). Empirical evidence associated with quite diverse
economies, and spanning different time periods, showed that the capital-intensity effect
overshadows the price effect, although there are cases where the latter effect is strong
enough that it can supersede the former giving rise to extrema and ‘price-labour value
reversals’ (also see Tsoulfidis and Mariolis, 2007, Tsoulfidis, 2008, Mariolis and
1 The terms ‘actual production prices’ and ‘actual profit rate’ are used to signify production prices and profit rate that correspond to the ‘actual’ real wage rate. The latter is estimated on the basis of the available input-output data.
5
Tsoulfidis, 2009). It then follows that the idea of representing the PPR curves through
linear or, a fortiori, quadratic approximations is absolutely justifiable and empirically
powerful (Bienenfeld, 1988; Shaikh, 2012; Iliadi et al., 2014).
(iv). Although the actual economies deviate considerably from the Ricardo-Marx-
Dmitriev-Samuelson ‘equal value compositions of capital’ case, the WPR curves are
near-linear, i.e. the correlation coefficients between the distributive variables tend to be
above 99%, and their second derivatives change sign no more than once or, very rarely,
twice, irrespective of the numéraire chosen (Leontief, 1985; Ochoa, 1989; Petrović,
1991; Han and Schefold, 2006).
All these findings imply that, although the actual economies cannot be analyzed
on the basis of ‘neoclassical parables’, the role of price-feedback effects is actually of
limited quantitative significance.
In the late-2000s the relevant research took a new direction on the basis of the
modern classical theory of value corollaries and the spectral representation (or, in more
general terms, the ‘modern state variable representation’) of linear systems.2 It has been
particularly pointed out that the functional expressions of the price-wage-profit rate
relationships admit lower and upper norm bounds, while their monotonicity could be
connected to the characteristic value distribution of the matrix of vertically integrated
technical coefficients and, therefore, to the ‘effective rank (or dimensions)’ of this
matrix. Since nothing can be said a priori about this crucial factor in real-world
economies, the examination of actual input-output data became absolutely necessary.
Thus, it has been well-ascertained that, across countries and over time, the moduli
of the eigenvalues as well as the singular values of actual economies follow
exponentially decaying trends. Moreover, when the capital stock matrices are taken into
account, they are characterized by a nearly ‘L-shaped’ pattern. Namely, in the latter,
more realistic case, the decay of the characteristic values is remarkably faster (see
Mariolis and Tsoulfidis, 2016b). This new stylized fact implies that only a few
eigenvalues really matter for the observed shapes of the P-WPR curves, which is another
way to say that these curves tend to be similar to those of low-dimensional systems. In
effect, it seems that matrix similarity transformations of the price system that result in
only a few industries extract the essential features contained in the original-actual
2 See Schefold (2008, 2013a), Mariolis and Tsoulfidis (2009, 2011, 2014b). For further analytical
investigations, see Schefold (2013b, 2016), Mariolis (2015a), Mariolis and Tsoulfidis (2016a), Shaikh
(2016, Chap. 9).
6
system and provide the basis for constructing reliable approximations of the observed
relationships.
The objective of this paper is to provide a unified treatment of both the theoretical
and empirical fundamentals of this recently developed research line that suggests – not
the irrelevance of Sraffian analysis but – a new logic approach for (i) revealing the
essential properties of the static and dynamic behaviour of a linear, closed and single-
product system as a whole; (ii) determining the extent to which these properties deviate
from those predicted by the traditional theories of value; and (iii) deriving meaningful
theoretical results consistent with the available empirical evidence. Supported also by
new empirical evidence, the present paper shows that the effective rank of actual
economies is rather low and, therefore, their price characteristic features tend to be
similar to those of “uncontrollable” (Kalman, 1961) and “irregular” (Schefold, 1971)
systems. Hence, actual economies may be described by just a few, or even a single,
‘hyper-basic’ industries without significant loss of information.
The remainder of the paper is structured as follows. Section 2 treats the theoretical
and empirical fundamentals of the new research line and, thus, points out the
uncontrollable-irregular features of actual economies. Section 3 provides new evidence
on the spectral properties of actual input-output structures using data from the US and
other major economies. Finally, Section 4 concludes the paper.
2. Spectral Decomposition of the Price System and Actual Economies
2.1. Preliminary relations
Let us suppose a linear circulating capital model of production described by the
irreducible n n matrix of direct technical coefficients, A, whose Perron-Frobenius
eigenvalue is less than one, and the surplus produced is distributed between profits and
wages. Let l be the 1 n vector of direct labour coefficients, w the uniform money
wage paid ex post, and r the economy-wide profit rate.3 On the basis of these
assumptions we can write the vector of production prices, p , as follows
3 The transpose of a 1 n vector [ ]jyy is denoted by
Ty . Furthermore, 1A denotes the Perron-
Frobenius eigenvalue of a semi-positive n n matrix [ ]ijaA , and T
1 1( , )A Ax y the corresponding
eigenvectors, while kA , 2,...,k n and 2 3 ... n A A A , denote the non-dominant
7
(1 )w r p l pA (1)
After rearrangement, equation (1) becomes
w r p v pH
or
w p v pJ (2)
where 1[ ] v l I A denotes the vector of vertically integrated labour coefficients, or
labour values, and 1[ ] H A I A the vertically integrated technical coefficients matrix.
Moreover, 1rR , 0 1 , denotes the relative profit rate, which equals the share of
profits in the SSS, and 1 1
1 11R A H the maximum possible profit rate (i.e. the profit
rate corresponding to 0w and p 0 ), which equals the ratio of the net product to the
means of production in the SSS (see Sraffa, 1960, pp. 21-23). Finally, RJ H denotes
the normalized vertically integrated technical coefficients matrix, 1 1 1R J H , and the
moduli of the normalized eigenvalues of system (2) are less than those of system (1), i.e.
1
1k k J A A holds for all k (see, e.g. Mariolis and Tsoulfidis, 2014b, pp. 213-214).
If SSC is chosen as the numéraire, i.e. T 1pz , where T T
1[ ] Az I A x and
T
1 1Alx , then the WPR curve is the following linear relation
1w (3)
and, if 1 ,
1 2 3(1 ) [ ] (1 ) [ ( ) ( ) ...] p v I J v I J J J (4)
which gives the production prices, expressed in terms of SSC, as polynomial functions
of .4 From equations (2), (3) and (4) it follows that:
(i) (0) 1w ;
(ii) (0) p v ;
(iii) (1) 0w ;
eigenvalues, and T( , )k kA Ax y
the corresponding eigenvectors. Finally, I denotes the n n identity
matrix, and je the j th unit vector.
4 If wages are paid ex ante, then the WPR curve is non-linear, i.e. 1(1 ) (1 )w R , and is no
greater than the share of profits in the SSS; however, equation (4) holds true. In the case of fixed capital à la Leontief (1953)-Bródy (1970), H
should be replaced by
1[ ]K I A , where K denotes the matrix of
capital stock coefficients.
8
(iv) (1)p is the left Perron-Frobenius eigenvector of J , expressed in terms of SSC, i.e.
T 1 T 1
1 1 1 1 1(1) ( ) ( [ ] ) J J J A Jp y z y y I A x y
or, since T T
1 1 1[ ] (1 ) A A AI A x x and matrices A and J have the same eigenvectors,
T 1
1 1 1 1(1) [(1 ) ] A J J Jp y x y (5)
(v) excluding the trivial and unrealistic case of equal value compositions of capital,
where (0) (1)p p and, therefore, prices are constant and equal to the labour values, as
well as the case of two-industry systems, where the PPR curves are necessarily
monotonic, changes in income distribution may activate complex capital revaluation
effects, which imply that the direction of relative price-movements cannot be known a
priori.
All traditional statements with respect to the exact price movements cannot, in
general, be extended beyond a world where (i) there are no produced means of
production; or (ii) there are produced means of production, while the profit rate on the
value of those means of production is zero; or, finally, (iii) that profit rate is positive,
while the economy produces one and only one, single or composite, commodity (see
Samuelson 1953-1954, pp. 17-19; Sraffa, 1960, Chap. 6; Salvadori and Steedman,
1985). Consequently, the conceptual and analytical difficulties of the traditional theories
of value and distribution arise from the existence of complex interindustry linkages in
the realistic case of production of commodities and positive profits by means of
commodities.
2.2. Turning to the outside world
It should, however, be taken into account that the empirical results usually give quasi-
linear price movements in terms of SSC. These finding could be explained by the shape
of the eigenvalue distribution: the eigenvalues of actual matrices J follow a rectangular
hyperbola-like distribution in the case of circulating capital, and a nearly L-shaped form
in the – more realistic – case of the presence of fixed capital stocks. In other words, the
stylized facts show that (i) the non-dominant eigenvalues of J are, as a statistical mean,
by far lower than 1; and (ii) the large gap between the second and dominant eigenvalues
of J allow pretty accurate approximations of the PRP trajectories through low order
spectral approximations (Mariolis and Tsoulfidis, 2016a, 2016b).
9
Thus, although the actual matrices J appear to have full rank, the particular
distribution of their eigenvalues gives, however, rise to an effective rank much lower
than the actual rank. It then follows that even an effective rank (or dimensionality) of 1
is sufficient for a satisfactory approximation to the PPR trajectories.5
2.3. Finding the ‘Archimedean point’ In order to zero in on this fundamental point, which is supported by the available
empirical evidence, we decompose matrix J to its ‘spectral representation’ (see, e.g.
Meyer 2001, 517-518)
T 1 T T 1 T
1 1 1 1
2
( ) ( )n
k k k k k
k
J J J J J J J J JJ y x x y y x x y (6)
If there are strong quasi-linear dependencies amongst the technical conditions of
production in all the vertically integrated industries, then [ ] 1rank J , or 0k J for all
k , and, therefore, equation (6) implies that A T 1 T
1 1 1 1( ) J J J JJ J y x x y . Hence, from
equation (4) it follows that
A A 1(1 ) (0)[ ] p p p I J
or, by applying the Sherman-Morrison formula,6
A 1 T 1 T
1 1 1 1(1 ) (0)[ (1 ) ( ) ] J J J Jp p p I y x x y
or, invoking equations (5) and T 1
1 1(0) (1 ) J Ap x ,
A (1 ) (0) (1) p p p p (7)
namely, Ap is a linear (‘convex’) combination of the extreme, economically significant,
values of the price vector, (0)p and (1)p .
This eigenvalue decomposition rank-one approximation for the price vector has
the following properties:
(i). It is linear and exact at the extreme values of .
(ii). Its accuracy is directly related to the magnitudes of 1
k J
.
(iii). When [ ] 1rank J , it becomes exact for all .
5 For the corresponding treatment of the WRP curves; see Mariolis (2015a, 2015b) and Mariolis and
Tsoulfidis (2016a, Chap. 5).
6 Let χ , ψ be arbitrary n vectors. Then T Tdet[ ] 1 I χ ψ ψχ
and, iff
T 1ψχ ,
T 1 T 1 T[ ] (1 ) I χ ψ I ψχ χ ψ (see, e.g. Meyer 2001, p. 124).
10
In that latter, ideal-type (in the Weberian sense) case, i.e. AJ J , the economy
exhibits the following two essential characteristics:
(i). Irrespective of the direction of the labour value vector (0) p v , it holds that
A T 1
1 1 1 1(0) (0)( ) [(1 ) ] (1)h h A J J Jp J p J y x y p , 1,2,...h
since
A T T 1 T A
1 1 1 1 1 1( ) ( ) ( )h h h J J J J J JJ y x y x x y J
Hence, the nxn ‘Krylov matrix’
T T T T 1 T T[ (0), (0),...,[ ] (0)]np J p J p
has rank equal to 2 and, therefore, the economy is said to be ‘irregular’ or, more
specifically, ‘regular of rank 2’.7 This means that the price vectors relative to any 3
distinct values of the profit rate ( 0 1 ) are linearly dependent (see Bidard and
Salvadori, 1995).
By contrast, an n economy is said to be ‘regular of rank n ’ or ‘completely
regular’ iff the aforementioned Krylov matrix has rank equal to n or, equivalently, iff
no right eigenvector of J is orthogonal to v . In that case, the price vectors relative to
any n distinct values of the profit rate are linearly independent. The concepts of
‘regularity/irregularity’ have been introduced by Schefold (1971), who argued that
irregular systems are not generic:
[T]he price vector of a [completely] regular Sraffa system is not only not constant, but
its variations in function of the rate of profit result in a complicated twisted curve
such that the n price vectors belonging to n different levels of the rate of profit […]
span a ( 1)n dimensional hyperplane which never contains the origin […]. [T]he
[completely] regular systems are the rule from a mathematical point of view […] the
set of irregular Sraffa systems with ( , )n n input-output-matrices is of measure zero
in the set of all Sraffa systems with the same number of commodities and industries.
But this observation taken by itself does not mean much. The set of all semi-positive
decomposable ( , )n n matrices is also of measure zero in the set of all semi-positive
( , )n n matrices, and yet it is quite clear that the analysis of the “exceptional”
decomposable matrices is of greatest economic interest, although they are more
difficult to handle than indecomposable matrices. There is an excellent economic
7 Obviously, in the theoretical case of equal value compositions of capital, the economy is regular of rank
1, irrespective of the rank of J .
11
reason why decomposable systems are important: pure consumption goods and other
non-basics exist; therefore decomposable systems exist. I should like to argue that
matters are quite different with irregular systems. I believe that there is no economic
reason why real systems should not be [completely] regular or why irregular systems
should exist in reality; irregularity is only a fluke, or, at best, an approximation.
(Schefold, 1976, p. 27)
In order to complete the picture, it is necessary to explicate that these concepts
are algebraically equivalent to those of ‘controllability/uncontrollability’ that have been
introduced by Kalman (1961) and apply to the following dynamic version of the price
system:
1t t tw p v p J , 0,1,...t
where denotes the exogenously given nominal relative profit rate, and 0 p 0
(Mariolis, 2003). Iff the aforementioned Krylov matrix has rank equal to n , then this
dynamic price system is said to be completely controllable, which means that the initial
state 0p can be transferred, by application of tw , to any state, in some finite time.
In our present case, i.e. AJ J , 1tp is a linear (‘conical’) combination of v and
1Jy , irrespective of the input sequence, tw , i.e.
1
1 0 1 1 1( ... )t t
t t tw w w w b Jp v y
where T 1 T
1 1 1( ) ( )b J J Jy x vx (compare with equation (7)). This uncontrollable system (or,
more generally, the low-rank controllable systems) seems to have some correspondence
with the ‘autopoietic systems’ of living and social systems theory (see, e.g. Dekkers,
Chap. 7).
(ii). The Schur triangularization theorem (see, e.g. Meyer, 2001, pp. 508-509) implies
that AJ can be transformed, via a semi-positive similarity matrix T , into
A
12 1 ( 1)A 1 A
( 1) 1 ( 1) ( 1)
1 [ ] n
n n n
JJ T J T
0 0 (8)
where the first column of T is T
1Jx , the remaining columns are arbitrary, and the vector
A
12J is necessarily positive (Mariolis, 2013). If, for instance,
T T T
1 2[ , ,..., ]n JT x e e (8a)
then
12
A T 1
12 1 1 2 1 3 1 1( ) [ , ,..., ]ny y y J J J J JJ y x
(8b)
The similarity matrix, T , defines a new coordinate system in which the original
system matrix, AJ J , is represented by a semi-positive triangular matrix, AJ , which
has the eigenvalues along its main diagonal. Thus, the original price system (2) is
decomposed as follows:
A 1( )w p v p TJ T
or, post-multiplying by 1T ,
A
w π ω πJ (9)
where π pT , ω vT denote the transformed vectors of price and labour values,
respectively, T
1 1 Jpx and T
1 1 Jvx . The first equation in the transformed price system
(9) corresponds to an industry producing a composite pure capital good, which is no
more than the SSS, whereas the remaining equations correspond to non-uniquely
determined industries producing pure consumption goods. It then follows that, even
when the matrix J is indecomposable, the original system is equivalent to an
economically significant and generalized (1 by 1n ) Marx-Fel’dman-Mahalanobis (or,
in more traditional terms, ‘corn-tractor’) system. Hence, the transformed industry
producing the pure capital good can be characterized as ‘hyper-basic’.
2.4. Matching the pieces
In the ideal-type case [ ] 1rank J the economy is regular-controllable of rank 2 and
economically equivalent to a decomposable system with one basic commodity and 1n
non-self-reproducing non-basics. Thus, on the one hand, the price side of the economy is
‘a little more’ complex than that of a pure labour theory of value economy and, at the
same time, much simpler than that of a completely regular-controllable economy. In
fact, its price side corresponds to that of the traditional neoclassical theory of value. On
the other hand, the economy can be fully described by a triangular matrix with only n
positive technical coefficients and, therefore, its production structure is ‘a little’ more
complex than that of ‘Austrian’-type economies, where the technical coefficients matrix
is, by assumption, strictly triangular (see, e.g. Burmeister, 1974).
When [ ] 2rank J , the spectral representation of the system matrix continues to be
a powerful tool for constructing higher-rank approximations of the PPR curves that may
13
involve more than one hyper-basic industry and, thus, various ideal-types for analyzing
the actual system.8
Finally, when, as is the case with actual economies, [ ]rank nJ but a particular
eigenvalue distribution gives rise to an effective rank much lower than the actual rank,
the economy tends to respond as an irregular-uncontrollable system. Consequently, the
real, after-Sraffa paradox, in the sense of knowledge vacuum, is not the ‘paradoxes in
capital theory’ but the very fact that, in correspondence to the rectangular hyperbola-like
distribution of their eigenvalues, actual economies constitute ‘almost irregular-
uncontrollable’ systems.
3. Empirical Evidence
Starting off with the distribution of eigenvalues for eight major economies, we provide
new empirical evidence that supports the previous analysis. We restrict ourselves to a
single year (2011) provided that the detected configuration of eigenvalues is pretty much
the same for all the actual economies that have been tested so far (Mariolis and
Tsoulfidis, 2016a, Chaps. 5-6, 2016b). We use data from the World Input-Output
Database (http://www.wiod.org), where the number of industries is not different across
economies and the data are compiled with the same methods and they are expressed in
dollars thereby facilitating inter-country comparisons (also see Timmer et al., 2015).
Table 1 reports the moduli of the eigenvalues of J (sorted in descending order) and
three metrics of distribution of moduli of the non-dominant eigenvalues, namely, (i) the
arithmetic mean, AM, that assigns equal weight to all moduli; (ii) the geometric mean,
GM, that assigns more weight to lower moduli, and, therefore, is more appropriate in
detecting the central tendency of an exponential set of numbers; and (iii) the so-called
spectral flatness, SF, defined as the ratio of the geometric mean to the arithmetic mean,
and shows how spiky or flat is the distribution under consideration.
8 Spectral higher-rank approximations can also be derived from the singular values of J , i.e. from the
square roots of the eigenvalues of the symmetric matrix TJJ . For the relationships between these spectral
approximations and Bienenfeld’s (1988) and Steedman’s (1999) polynomial approximations, see Mariolis and Tsoulfidis (2016a, Chap. 5).
14
Table 1. Distribution of the moduli of eigenvalues; Australia, Brazil, P.R. China, France,
Germany, India, Japan, and USA, year 20119
Eigenvalues Ranking
AUS
BRZ
CHN
FRC
GER
IND
JPN
USA
1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2 0.359 0.379 0.409 0.437 0.526 0.408 0.472 0.488
3 0.289 0.359 0.316 0.333 0.399 0.408 0.472 0.488
4 0.240 0.311 0.291 0.292 0.399 0.264 0.422 0.429
5 0.212 0.254 0.247 0.257 0.348 0.245 0.406 0.293
6 0.184 0.254 0.221 0.224 0.260 0.232 0.272 0.293
7 0.184 0.230 0.220 0.224 0.218 0.232 0.251 0.235
8 0.154 0.230 0.220 0.221 0.218 0.167 0.230 0.207
9 0.136 0.216 0.208 0.221 0.209 0.122 0.214 0.207
10 0.136 0.216 0.194 0.193 0.209 0.122 0.155 0.147
11 0.115 0.174 0.099 0.184 0.180 0.122 0.155 0.117
12 0.110 0.162 0.084 0.157 0.180 0.122 0.121 0.106
13 0.104 0.103 0.084 0.152 0.171 0.096 0.121 0.105
14 0.104 0.103 0.067 0.152 0.171 0.096 0.076 0.105
15 0.088 0.097 0.058 0.122 0.150 0.085 0.063 0.085
16 0.088 0.050 0.039 0.122 0.124 0.063 0.063 0.085
17 0.067 0.050 0.039 0.109 0.112 0.054 0.049 0.077
18 0.067 0.024 0.035 0.082 0.112 0.054 0.037 0.077
19 0.058 0.018 0.035 0.075 0.093 0.028 0.037 0.051
20 0.045 0.016 0.025 0.068 0.093 0.026 0.036 0.051
21 0.043 0.016 0.019 0.068 0.091 0.026 0.036 0.049
22 0.043 0.014 0.014 0.056 0.069 0.014 0.027 0.049
23 0.039 0.014 0.014 0.042 0.042 0.014 0.017 0.049
24 0.039 0.003 0.012 0.035 0.034 0.014 0.017 0.032
25 0.022 0.003 0.012 0.032 0.034 0.009 0.016 0.032
26 0.020 0.000 0.010 0.024 0.031 0.009 0.016 0.024
27 0.013 0.000 0.010 0.024 0.031 0.005 0.012 0.024
28 0.013 0.000 0.005 0.019 0.030 0.001 0.012 0.023
29 0.011 0.000 0.005 0.008 0.030 0.001 0.012 0.017
30 0.005 0.000 0.004 0.008 0.025 0.001 0.009 0.008
31 0.002 0.000 0.004 0.008 0.015 0.000 0.008 0.005
32 0.002 0.000 0.003 0.002 0.007 0.000 0.006 0.003
33 0.000 0.000 0.000 0.002 0.005 0.000 0.006 0.001
34 0.000 0.000 --- 0.000 0.005 0.000 0.000 0.001
AM 0.117 0.126 0.121 0.146 0.165 0.119 0.143 0.146
GM 0.024 0.001 0.022 0.060 0.085 0.012 0.048 0.058
SF 0.201 0.006 0.179 0.415 0.517 0.099 0.339 0.397
9 China’s industry 19 (i.e. “Sale, Maintenance and Repair of Motor Vehicles and Motorcycles; Retail Sale
of Fuel” contains no data; thus, the number of eigenvalues for this economy is 33. It is also noted that the
last eigenvalues for India and Brazil are almost indistinguishable from zero, and this results in smaller
geometric means and spectral flatness.
15
From these findings and the analytical numerical results it follows that:
(i). The moduli of the first non-dominant eigenvalues fall markedly, whereas the rest
constellate in much lower values forming a ‘long tail’, and the empirical evidence so far
suggests that they would not play any significant role in the observed shapes of P-WPR
curves of the economies. After experimentation with various possible functional forms,
we found that a single exponential functional form fits all the moduli data pretty well, as
this can be judged by the high R-square and the fact that all the estimated coefficients
are statistically significant, with zero probability values. This form is displayed in Figure
1 and given by10
0.2
0 1 exp( )y x , 0 0 and 1 0
where y stands for the moduli of the eigenvalues (displayed on the vertical axis for each
of our eight countries in Figure 1), whereas x stands for the respective ranking of the
moduli of eigenvalues (displayed on the horizontal axis), and 0 , 1 are parameters to
be estimated. Finally, the regression equations are displayed inside the graphs for each
of the countries under examination. This functional form is surprisingly similar to that
associated with the findings of previous studies for a number of diverse economies and
years quite distant from each other (see Mariolis and Tsoulfidis, 2016a, Chaps. 5-6,
2016b).
(ii). The complex (as well as the negative) eigenvalues tend to appear in the lower ranks,
i.e. their modulus is relatively small. However, even in the cases that they appear in the
higher ranks, i.e. second or third rank, the real part has been found to be much larger
than the imaginary part, which is equivalent to saying that the imaginary part may even
be ignored. Moreover, in the fewer cases that the imaginary part of an eigenvalue
exceeds the real one, not only their ratio is relatively small but also the modulus of the
eigenvalue can be considered as a negligible quantity. Finally, by inspecting all of our
eigenvalues, we observe that, in general, the imaginary part gets progressively smaller.
Consequently, the already detected distributions of the moduli can be viewed as fair
representation of the distributions of the eigenvalues, and the complex eigenvalues play
no perceptible role in the question at hand (however, they may be crucial in other topics;
see, e.g. Rodousakis, 2012, 2016).
10 In fact, we tried an optimization procedure to find the best possible form, and from the many
possibilities, we opted for a simple but, at the same time, general enough to fit the moduli of the
eigenvalues of all economies under consideration.
16
Figure 1. Exponential fit of the distribution of the moduli of the eigenvalues; Australia, Brazil,
P.R. China, France, Germany, India, Japan, and USA, year 2011
17
By focusing on the US economy, the general picture remains the same for the
much larger in dimensions input-output tables, which have been published by the
Bureau of Economic Analysis (http://www.bea.gov), for the benchmark years 1997
( 488n ), 2002 ( 426n ) and 2007 ( 389n ). Figure 2 displays the location of the
eigenvalues in the complex plane for the year 2007, while Figure 3 displays the
exponential fit of the distribution of the moduli of the eigenvalues (the axes are as in
Figure 1).11 A visual inspection of eigenvalues displayed in Figure 2 makes it
abundantly clear that the majority of the non-dominant eigenvalues are crowded at very
low values and bounded in a relatively small region of the unit circle.
1.0 0.5 0.5 1.0
1.0
0.5
0.5
1.0
Figure 2. The location of the eigenvalues in the complex plane; USA, year 2007, 389n
11 These results are rather similar to those for the years 1997 and 2002, that is 0 0.754 , 1 0.641 ,
2 0.3 , 2 98.7%R (year 1997) and 0 0.800 , 1 0.678 , 2 0.3 ,
2 99.1%R (year
2002); see Mariolis and Tsoulfidis (2014b, pp. 216 and 218).
18
Figure 3. Exponential fit of the moduli of the eigenvalues; USA, year 2007, 389n
Although the larger dimensions input-output tables provide the basis for a more
detailed analysis of the P-WPR curves, the hitherto empirical evidence suggests,
however, that they do not lead to essentially different conclusions regarding the features
of the eigenvalue distributions and, therefore, the shape of those curves.12 Thus, our next
focus is on the data of the last available 15 x 15 input-output table of the US economy
for the year 2014 (https://www.bea.gov/industry/io_annual.htm; provided that for the
other years the results are not very different).13 Figure 4 displays the exponential fit of
the distribution of the moduli of the eigenvalues (the axes are as in Figure1), and Figure
5 displays the 15 trajectories of the production price-labour value ratios, expressed in
terms of SSC and measured on the vertical axis, as functions of the relative profit rate,
0 1 , which is measured on the horizontal axis (see equations (4) and (5)).
12 See Mariolis and Tsoulfidis (2011, pp. 105-109, 2014b, pp. 215-219); Gurgul and Wójtowicz, (2015);
Pires and Shaikh (2015) and Shaikh (2016, Chap. 9). 13 For the nomenclature of the industries, see Table A.1 in the Appendix.
19
Figure 4. Exponential fit of the moduli of the eigenvalues; USA, year 2014, 15n
Figure 5. Production price-labour value ratios as functions of the relative profit rate; USA, year
2014, 15n
From these results, which are in close accord with those of all past studies on this
topic, it follows that:
(i). At first sight, the trajectories of prices appear that pretty much move monotonically
(Figure 5). Thus, the capital-intensity effects overshadow the price effects, and this
finding seems to match the underlying eigenvalue distribution (Figure 4).
20
(ii). However, a more detailed examination reveals that in two out of fifteen industries
(or 13.3% of the cases under consideration) there is price-labour value reversals, that is,
the production price-labour value curves associated with the vertically integrated
industries 2 and 9 display maxima and then cross the line of equality between production
prices and labour values at a positive value (and less than unity) of the relative profit
rate. As the left-hand side panel of Figure 6 shows (where we restricted the relative
profit rate up to 0.30 for reasons of visual clarity), the line of price-value equality is
crossed at a relative profit rate in the range of 0.15 to 0.20 (industry 2) or 0.20 to 0.25
(industry 9). This phenomenon is reflected in the right-hand side panel of Figure 6,
which depicts the relevant capital-intensities (measured on the vertical axis) and
indicates that, as the relative profit rate increases, these two industries are transformed to
labour intensive relative to the SSS.14
(a) Production price-labour value ratios (b) Capital-intensities
Figure 6. (a) Production price-labour value ratios; and (b) capital-intensities of vertically
integrated industries 2 and 9, as functions of the relative profit rate; USA, year 2014, 15
industries
14 The capital-intensity of the vertically integrated industry producing commodity j is estimated by
T 1j jv
pHe , while the capital-intensity of the SSS equals the reciprocal of the Standard ratio, 1
1R H ,
which in our case is approximately equal to 0.959.
21
(iii). There are good grounds to test the empirical validity of the rank-1 approximation
for the production price vector, which is based on matrix A T 1 T
1 1 1 1( ) J J J JJ y x x y and
defined by equation (7). Since this approximation is linear and exact at the extreme,
economically significant, values of the relative profit rate, its absolute errors, A
j jp p ,
are maximized at intermediate values of this distributive variable. Indeed, as Figure 7
shows, in the negative direction, the maximal error is almost 0.103 at 0.60 (and
associated with industry 1) and, in the positive direction, the maximal error is 0.089 at
0.60 (and associated with industry 10). Moreover, the ‘average absolute deviation’,
without the extreme values of , is 2.67% . Thus, it can be concluded that, although
higher-rank approximations would lead to more accurate price curves, the linear
approximation works well.
22
Figure 7. The absolute errors in the linear approximation of production prices as
functions of the relative profit rate
Having established that AJ is a good first approximation of J , in the sense that
both matrices give rise to price trajectories close to each other, we apply the similarity
transformation, defined by equations (8) and (8a, b), to these matrices. It then follows
that:
(i). The first row of the triangular, semi-positive and rank-1, matrix
A 1 A 1[ ] J T J T T J E T
where T 1 T
2
( )n
k k k k k
k
J J J J JE y x x y denotes the error matrix (see equation (6)), is:
1 0.139 0.136 0.230 0.385
0.114 0.123 0.266 0.182 0.097
0.127 0.151 0.191 0.147 0.158
(ii). The first row of the non-triangular, non-semi-positive and rank-15, matrix
1J T JT is:
1 0.124 0.121 0.204 0.343
0.101 0.110 0.237 0.162 0.086
0.113 0.134 0.170 0.131 0.140
23
(iii). The ‘mean absolute percentage deviation’ between these two rows is 11.5%,
whereas the ‘normalized d distance’ (Steedman and Tomkins, 1998; Mariolis and
Soklis, 2010, p. 94) is 2.3%.
Since, on the one hand, AJ is a good approximation of J and, on the other hand,
the first row of AJ is a good approximation of the first row of J , it follows that the
essential PRP-information embedded in the original-actual economy is captured by the
first row of J and extracted in the first row of AJ . Hence, the economy defined by the
rank-1 matrix AJ and producing a pure capital good and 1n pure consumption goods,
can be considered as a fairly representative simulacrum of the actual economy.
It goes without saying that, due to the nearly L-shaped form of the eigenvalue
distribution, such a representation would be, in general, much more powerful in the
presence of fixed capital stocks.
4. Concluding Remarks
The input-output data of actual single-product economies suggested that the majority of
the non-dominant eigenvalues of the normalized vertically integrated technical
coefficients matrices are crowded at very lοw values and bounded in a relatively small
region of the unit circle. This stylized fact implies that the economically relevant
movements of prices in function of the profit rate do not follow erratic patterns and,
therefore, their approximation through low-order formulae (ranging from linear to
quadratic) give accurate results, which can be improved only marginally by the inclusion
of higher order terms. Consequently, actual economies constitute almost irregular-
uncontrollable systems, which can be adequately described by a single or just a few
hyper-basic industries.
The theoretical and empirical findings indicate that many of issues still lie hidden
underneath the surface of capital theory debates and may be discovered through a
combination of proper economic theory and use of data derived from the structure of
actual economies. In such a direction, it appears that, although a lot is lost by one-
commodity world postulations, embedded, explicitly or otherwise, in the traditional
theories of value, there is room for using models with at most three basic commodities
and many non-self-reproducing non-basics as surrogates for actual single-product
24
systems. Given that little is gained by considering higher dimensions, it can be boldly
stated that: less is more.
Future research work should (i) seek for possible relationships between measures
of almost irregularity and monotonicity of price curves; (ii) delve into the proximate
determinants of the irregular-uncontrollable aspects of actual economies; and (iii)
incorporate the cases of many primary inputs (such as non-competitive imports) and
joint-products.
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Appendix
The 15 industries input-output structure of the USA is reported in Table Α.1. For the
estimation procedures we refer to Mariolis and Tsoulfidis (2016a, Chap. 3) and the
literature therein.
Table A.1. Nomenclature of 15 Industries, USA economy
1 Agriculture, forestry, fishing, and hunting
2 Mining
3 Utilities
4 Construction
5 Manufacturing
6 Wholesale trade
7 Retail trade
8 Transportation and warehousing
9 Information
10 Finance, insurance, real estate, rental, and leasing
11 Professional and business services
12 Educational services, health care, and social assistance
13 Arts, entertainment, recreation, accommodation, and food services
14 Other services, except government
15 Government